Formula:KLS:14.22:04

<< Formula:KLS:14.22:03

formula in q-Bessel

Formula:KLS:14.22:05 >>

${\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{a^{k}}{\left(q;q\right)_{k}}q^{\binomial{k+1}{2}}y_{m}\!\left(q^{k};a;q\right)y_{n}\!\left(q^{k};a;q\right){}=\left(q;q\right)_{n}\left(-aq^{n};q\right)_{\infty}\frac{a^{n}q^{\binomial{n+1}{2}}}{(1+aq^{2n})}\,\delta_{m,n}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle a>0}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\Sigma}}}$ : sum : http://drmf.wmflabs.org/wiki/Definition:sum
${\displaystyle{\displaystyle{\displaystyle(a;q)_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Pochhammer symbol : http://dlmf.nist.gov/5.18#i http://dlmf.nist.gov/17.2#SS1.p1
${\displaystyle{\displaystyle{\displaystyle\genfrac{(}{)}{0.0pt}{}{n}{k}}}}$ : binomial coefficient : http://dlmf.nist.gov/1.2#E1 http://dlmf.nist.gov/26.3#SS1.p1
${\displaystyle{\displaystyle{\displaystyle y_{n}}}}$ : ${\displaystyle{\displaystyle{\displaystyle q}}}$-Bessel polynomial : http://drmf.wmflabs.org/wiki/Definition:qBessel
${\displaystyle{\displaystyle{\displaystyle\delta_{m,n}}}}$ : Kronecker delta : http://dlmf.nist.gov/front/introduction#Sx4.p1.t1.r4

Bibliography

Equation in Section 14.22 of KLS.

URL links

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<< Formula:KLS:14.22:03

formula in q-Bessel

Formula:KLS:14.22:05 >>